Beyond coherence : recovering structured time-frequency representations

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Beyond coherence : recovering structured time-frequency representations

Lasse Borup, Rémi Gribonval, Morten Nielsen

To cite this version:

Lasse Borup, Rémi Gribonval, Morten Nielsen. Beyond coherence : recovering structured time- frequency representations. [Research Report] PI 1833, 2007, pp.13. �inria-00130123v2�

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CAMPUS UNIVERSITAIRE DE BEAULIEU - 35042 RENNES CEDEX - FRANCE

ISSN 1166-8687

1833

BEYOND COHERENCE : RECOVERING STRUCTURED TIME-FREQUENCY REPRESENTATIONS.

LASSE BORUP , RÉMI GRIBONVAL , MORTEN NIELSEN

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INSTITUT DERECHERCHE EN INFORMATIQUE ETSYSTÈMESALÉATOIRES

Campus de Beaulieu – 35042 Rennes Cedex – France Tél. : (33) 02 99 84 71 00 – Fax : (33) 02 99 84 71 71 http://www.irisa.fr

Beyond coherence : recovering structured time-frequency representations.

Lasse Borup* , R´emi Gribonval ** , Morten Nielsen ***

Syst`emes cognitifs Projet Metiss

Publication interne n˚1833 — F´evrier 2007 — 14 pages

Abstract: We consider the problem of recovering a structured sparse representation of a signal in an overcomplete time-frequency dictionary with a particular structure. For infinite dictionaries that are the union of a nice wavelet basis and a Wilson basis, sufficient conditions are given for the Basis Pursuit and (Orthogonal) Matching Pursuit algorithms to recover a structured representation of an admissible signal. The sufficient conditions take into account the structure of the wavelet/Wilson dictionary and allow very large (even infinite) support sets to be recovered even though the dictionary is highly coherent.

Key-words: sparse representation, overcomplete dictionary, matching pursuit, basis pursuit, time-frequency dictionaries, wavelets, Wilson basis, linear programming, greedy algorithm, , identifiability, inverse problem.

(R´esum´e : tsvp)

This paper has been submitted for possible publication to Applied and Computational Harmonic Analysis. This work is supported in part by the European Union’s Human Potential Programme, under contract HPRN-CT-2002-00285 (HASSIP).

*Department of Mathematical Sciences, Aalborg University, Fr. Bajers Vej 7G, DK-9220 Aalborg East, Denmark, lasse@math.aau.dk

**remi.gribonval@irisa.fr.

***Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7 G, DK-9220 Aalborg East, Denmark, mnielsen@math.wustl.edu

Centre National de la Recherche Scientifique Institut National de Recherche en Informatique (UMR6074) Université de Rennes 1 – Insa de Rennes et en Automatique – unité de recherche de Rennes

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Sur l’identification de représentations temps-fréquence parcimonieuses structurées

Résumé : Nous nous intéressons au comportement des principaux algorithmes d’approximation parcimonieuse sur des signaux admettant une représentation à la fois parcimonieuse et structurée dans des dictionnaires temps-fréquence particuliers constitués de l’union d’une base d’ondelette et d’une base de Wilson, qui sont utilisés pour certains schémas de codage audio dits hybrides. Il s’agit de déterminer des conditions de ”structure” suffisantes garantissant qu’une représentations d’un signal dans ce type de dictionnaire soit retrouvée par les algorithmes de type Matching Pursuit (MP) etBasis Pursuit.

Les conditions suffisantes les plus classiques, basées sur la cohérence (cumulative) du dictionnaire, sont en effet inopérantes dans ce contexte car elles amènent à se restreindre à des signaux qui coincident avec un atome du dictionnaire, et les résultats probabilistes valables pour des dictionaires aléatoires ne peuvent être directement appliqués sur un tel dictionnaire très structuré. En supposant que le dictionaire est généré à partir de fonctions (ondelette mère et fenêtre d’analyse) suffisament régulières, nous exprimons ici des conditions suffisantes beaucoup moins pessimistes qui prennent en compte la structure particulière du dictionnaire hybride ondelettes/Wilson. Les conditions obtenues reposent sur le fait que ce dictionnaire peut être décomposé en sous-ensembles d’ondelettes et de fonctions de base de Wilson approximativement associés à des ”blocs” temps-fréquence disjoints, et qui forment essentiellement l’union de deux bases maximalement incohérentes des fonctions ayant ces blocs pour support.

Mots clés : représentation parcimonieuse, dictionnaire redondant, matching pursuit, basis pursuit, dictionnaire temps-fréquence, ondelette, base de Wilson, programmation linéaire, algorithme glouton, séparation de sources, traitement du signal multicanal, iden- tifiabilité, problème inverse.

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Beyond coherence : recovering structured time-frequency representations 3

Contents

1 Introduction 4

2 Wavelet and local cosine dictionary 6

2.1 Basis functions . . . . 6

2.2 Cumulative coherence . . . . 7

2.3 Sketch with time-frequency blocks . . . . 7

2.4 Inner products between wavelet and local cosine basis functions . . . . 8

2.5 Upper bound on the cumulative coherence . . . . 9

3 Conclusion 11

A Time-frequency estimates 12

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4 Borup, Gribonval & Nielsen

1 Introduction

Let Φ = [g_i]_i∈F be an at most countable collection of normalized elements in a Hilbert spaceH. We say thatΦis a besselian dictionary if the associated linear mapΦ:ℓ²(F)→ H given by Φ[(ck)k] = P

i∈F ckg_i is bounded. In this note we consider the problem of recovering a sparse representation S

X =Φ(S), S ∈ℓ²(F), (1) of a signal X ∈ H relative to a besselian dictionary Φ with a specific structure. Sparse representations provide a very useful tool to solve many problems in signal processing including blind source separation, feature extraction and classification, denoising, and detection, to name only a few (see also [13], and references therein). Several algorithms, such as Basis Pursuit (ℓ¹-minimization) and Matching Pursuits (also known as greedy algorithms), have been introduced to compute sparse representations/approximations of signals. The problem we face is that such algorithms a priori only provide sub-optimal solutions. That is, we do get a representation of the type (1), but we may not recover the sparse representation S of X.

Several recent papers [5, 6, 7, 17, 10, 12] have identified situations where algorithms such as Basis Pursuit actually compute an optimal representation of a given signal, in the sense that they solve the best approximation problem under a constraint on the size of the support of the signal. Typically, one calculates the coherence of Φ

µ(Φ) = sup

i6=j |hg_i,g_ji|.

Then for signals X with a representation X =Φ(S) satisfying |supp(S)|<⌊¹₂(1 + 1/µ)⌋, Basis Pursuit will recover the representation S. One serious problem with this type of results using the coherence is that they represent worst caseestimates. For example, the coherence is close to one as soon as we have one pair of atoms that are approximately colinear while the rest of the dictionary may be much nicer. A more refined type of result can be obtained by considering the cumulative coherence introduced by Tropp [17]

µ1(Φ, m) := sup

|Λ|=m

sup

j6∈Λ

X

i∈Λ

|hg_i,g_ji|.

However, the cumulative coherence also gives a worst case estimate that does not take into account the finer structure of the dictionary, and the mentioned bounds are too weak for many applications.

One way to overcome these shortcommings is by shifting to a probabilistic viewpoint and consider random dictionaries. The probabilistic approach has been considered in a number of recent papers, see e.g. [1, 4]. Random dictionaries are typically created by picking a number of unit vectors randomly from some larger ensemble. The results on sparse representations using random dictionaries are typically much better than the cor- responding deterministic results. One problem is that the results are difficult to intrepret when we consider a specific dictionary.

We follow a different deterministic path in this note. The goal is to give more optimistic results for some concrete dictionaries that are often used in signal processing and harmonic analysis. The idea is to take into account some of the internal structure of the dictionary.

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The typical example of an admissible dictionary is the union of a nice wavelet basis and a Wilson basis.

The main tool to extend the classical estimates is to consider the setwise p-Babel function , which trivially extends the setwise Babel function defined in [11] as follows.

For 1 ≤p <∞ and a set I ⊆F we define µp(Φ, I) :=

sup

i /∈I

X

j∈I

|hg_i,g_ji|^p 1/p

. (2)

For S a family of subsets of F, we define thestructured p-Babel function as µp(Φ,S) := sup

I∈S

µp(Φ, I). (3)

Notice that we allow infinite dictionaries Φ so it may happen that µp(Φ, I) = +∞. The structured 1-Babel function µ1(Φ,S) generalizes the Babel functionµ1(Φ, m). In fact, let S^m ={I ⊆F :|I|=m}, m = 1,2, . . . ,|F|. Then µ1(Φ, m) =µ1(Φ,S^m). The case p= 1 is especially interesting due to the following result considered by Tropp [17] for the Babel function. The proof of Lemma 1 is a straightforward generalization of the proof in [17]

and was pointed out in [11]. It involves the sub-dictionary Φ_I = [gi]i∈I made of atoms from the set I ⊆F.

Lemma 1. Let X =Φ(S) and suppose that supp(S) = I is such that µ1(Φ, I) + max

ℓ∈I µ1(Φ_I, I\{ℓ})<1. (4)

Then Basis Pursuit and Orthogonal Matching Pursuit exactly recover the representation S of X.

In the special case where the nonzero coefficients in the representationS have similar magnitudes|ck| ≈cst, the condition (4) is also sufficient to ensure that simple thresholding will recover S [14, 16]. A similar condition involving the 2-Babel function instead of the 1-Babel function was recently shown to be related [9] to the probabilty of recovery with thresholding for simultaneous (multichannel) sparse approximation. All recovery results, which are expressed here for noiseless models X = Φ(S), have been shown to be stable to noise.

To illustrate how we can use dictionary structure to get improved recovery results, let us consider two examples. The first example is finite dimensional. InC^N we considerΦ_{F D} given as the union of the Dirac and the Fourier orthonormal basis for C^N. One easily checks that the dictionary is maximally incoherent with µ(ΦF D) = 1/√

N. Thus, the classical result shows that Basis pursuit recovers sparse signals having a a representation with support less than ⌊¹₂(1 +√

N)⌋atoms. Here we only estimate the size of the support.

The second example is infinite dimensional. We considerΦ_HW defined as the union of the Haar system (see e.g. [15])B1 ={h_n}^∞n=0and the Walsh system (see [8])B2 ={W_n}^∞n=0

on [0,1]. The Haar system and the Walsh system both form an orthonormal basis for L²([0,1]) soΦ_HW is a tight frame forL²([0,1]). The point we wish to make is the following;

it is easy to check thatµ(Φ_HW) = 1, so naivelyone would expect that no decent recovery result is possible. However, Φ_HW has a lot of structure we can exploit. In the time- frequency plane, the Walsh function Wn is supported in the block [0,1]×[n, n+ 1] while

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the Haar function hn with 2^j ≤n <2^j+1 is supported on Hj,0 := [0,1]×[2^j,2^j+1−1]. In fact, let Q(j) ={hk}²_k=2^j+1^j⁻¹ and Q^′(j) ={Wk}²_k=2^j+1^j⁻¹. Then Wj := spanQ(j) = spanQ^′(j) with Wj⊥Wj^′ for j 6=j^′, see [8]. Moreover, the 2^j-dimensional subdictionary Φ_HW(j) :=

Q(j)∪Q^′(j) is perfectly incoherent with µ(Φ_HW(j)) = 2^−j/2. Hence, if a signal x has a representation

x=c0+ X∞

j=0 2X^j−1

k=0

cj,kh2^j+k+dj,kW2^j+k

,

with |supp({c_j,k}k)|+|supp({d_j,k}k)|< ¹₂(1 + 2^j/2) for j ≥0 (the notation supp() stands for the support set where a sequence is nonzero, and | · | denotes the cardinality of such a set), then we can use the simple finite dimensional estimate using the coherence (and the fact thatW_j⊥W_j^′ forj 6=j^′) to conclude that MP and BP recover this representation of x. Notice how this estimate takes into account the structure of the dictionary and not only the size of the support of signals.

The main result of this paper is to extend the straightforward considerations forΦ_HW to other dictionaries with the same type of underlying structure. However, we will not assume that the dictionary can be decomposed into orthogonal finite dimensional dictionaries which will give rise to some added technicalities in the estimates. Our result holds for unions of an orthonormal wavelet basis {ψj,k} and a Wilson basis {gn,m} with sufficient smoothness, a type of dictionary which was proposed for audio signal modeling and compression by Daudet and Torr´esani [3]. For any pair c := cj,n and d := dn,m of coefficient sequences we define

Nj(c,d) := sup

n∈Z

max |supp({c_j,2^j_n+ℓ}²ℓ=0^j⁻¹)|,|supp({d_n,2^j_+ℓ}²ℓ=0^j⁻¹)| .

Theorem 1. There is a constant K (which depends on the support size and smoothness of the mother wavelet ψ and Wilson window function g) such that any pair of sequences

satisfying X

j≥0

Nj(c,d)·2^−j/2 < K

will be recovered by both Basis Pursuit and (Orthonormal) Matching Pursuit performed on the signal x=P∞

j=0

P

n∈Z

P2^j−1

ℓ=0 cj,2^jn+ℓψj,2^jn+ℓ+dn,2^j+ℓgn,2^j+ℓ

.

2 Wavelet and local cosine dictionary

In this section we introduce the main function dictionary considered in this paper. The dictionary is the union of an orthonormal wavelet and local cosine basis and is conse- quently a tight frame with frame constant 2. We will not discuss the details involved in the construction of these bases here, but just refer the reader to e.g. [15]. To avoid unnecessary technicalities, we only consider the univariate case.

2.1 Basis functions

Let φ and ψ be a scaling function and a wavelet, both with compact support, such that B¹ :={φk}k∈Z∪ {ψj,k}j≥0,k∈Z, is an orthonormal wavelet basis forL²(R), where

ψj,k(x) := 2^j/2ψ 2^jx−k

and φk(x) :=φ(x−k).

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Suppose that g is a smooth compactly supported “cut-off” function, and let gn,m(x) :=√

2g(x−n) cos(π(m+ ¹₂)(x−n))

for n ∈Z and m∈N₀ :=N∪ {0}. For a suitable choice ofg, B² :={g_n,m}n∈Z,m∈N₀ is an orthonormal basis forL²(R), see [15, Sec. 1.4]. The besselian dictionary considered is the tight frame Φ= B¹∪ B². It is indexed by F = F^φ∪F^ψ ∪F^g where F^φ (resp. F^ψ, F^g) indexes scaling functions (respectively wavelets, local cosines).

2.2 Cumulative coherence

We may partition any index set I ⊂F into scaling function indices I^φ =I∩F^φ, wavelet indices I^ψ =I ∩F^ψ and local cosine indices I^g =I∩F^g. Since each basis is orthogonal, the p-cumulative coherence of I is given by

µ^p_p(Φ, I) = max



 sup

ψ^′∈(F^φ∪F^ψ)\I



 X

ψ∈I^φ∪I^ψ

|hψ^′, ψi|^p+X

g∈I^g

|hψ^′, gi|^p



,

sup

g^′∈F^g\I



 X

ψ∈I^φ∪I^ψ

|hg^′, ψi|^p+X

g∈I^g

|hg^′, gi|^p









≤max



 sup

ψ^′∈F^φ∪F^ψ

X

g∈I^g

|hψ^′, gi|^p, sup

g^′∈F^g

X

ψ∈I^φ∪I^ψ

|hg^′, ψi|^p



 (5) where we slightly abused notations by confusing basis functions with their indices, e.g., in the notation g ∈I^g.

2.3 Sketch with time-frequency blocks

To estimate each of the two terms which appear in the maximum (5) we will partition further the index sets I^φ, I^ψ and I^g. Given j ∈ N₀ and n ∈ Z, it is easy to see that for a nice mother wavelet ψ, the 2^j functions {ψj,k}2^jn≤k<2^j(n+1) are essentially localized in time in the neighborhood of the interval [n, n+ 1], and essentially localized in frequency in the neighborhood of the interval [2^j,2^j+1]. In other words, they are localized around the “time-frequency block” Hj,n := [n, n+ 1] × [2^j,2^j+1]. The same goes for the 2^j local cosine functions {gn,m}2^j≤m<2^j+1, if g is well-localized in time and frequency. The coherence between any wavelet and Wilson function “living” on such a block is of the order 2^−j/2. The distinct regions{Hj,n}^n∈^Z^,j∈^N⁰ essentially¹ tile the time-frequency plane, and in contrast to the relatively large coherence between functions from the same tile of the partition, the coherence between any two functions in two different pieces is (much) smaller. Thus, we may cut the sets I^ψ and I^g into pieces in parallel to the tiling of the time-frequency plane, and define for j ≥0 and n ∈Z

I_j,n^ψ := {ψj,k ∈I^ψ, 2^jn≤k < 2^j(n+ 1)} I_j,n^g := {gn,m∈I^g, 2^j ≤m <2^j+1}.

1For a complete tiling one would also need to include low frequency regions [n, n+ 1]×[0,1]

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For a given wavelet ψj,k ∈/ I^ψ, letting n := ⌊2^−jk⌋ be such that 2^jn ≤ k < 2^j(n+ 1) we have

X

g∈I^g

|hψ_j,k, gi|^p =X

j^′≥0

X

n^′∈Z

X

g∈I^g

j′,n′

|hψ_j,k, gi|^p ≈ X

g∈I^g_j,n

|hψ_j,k, gi|^p ≈ |I_j,n^g | ·2^−jp/2 (6)

provided that the above sketchy analysis is valid. A similar estimate holds if the role of the wavelet and local cosine bases is exchanged, and the numbers

Nj(I) := sup

n∈Z

max

|I_j,n^ψ |,|I_j,n^g |

(7) are therefore involved in the estimation of the coherence. Indeed, if we assume that the index set I does not contain any scaling function (i.e., I^φ = ∅) and no Wilson function gn,0 either, we get the following estimate of the p-cumulative coherence

µp(Φ, I)≈sup

j≥0

2^−j/2Nj(I)^1/p

. (8)

The above approach is only a sketch: in practice the inner products between wavelets and local cosine functions do not depend as sharply as depicted here on the time-frequency block to which their indices belong, and we will see below how the above approach should be corrected.

Note that we assumed in this sketch that the index set I did not contain any low- frequency atom (i.e., no scaling function and no Wilson function of the type gn,0). This restriction is only natural since scaling functions and Wilson functions of this type have very similar shapes, and have a very large coherence, so if they were to be included in I then the p-cumulative coherence would almost certainly exceed one.

2.4 Inner products between wavelet and local cosine basis functions

To estimate the p-coherence from above we will need to control the inner products

|hψj,k, gn,mi| between wavelets and local cosine functions. The following lemmata give the fundamental estimates and are proved in Appendix A.

Lemma 2. Let φ, ψ, and g be three univariate functions such that for some C < ∞ and A >1 we have

max(|φ(ξ)b |,|ψ(ξ)b |,|bg(ξ)|)≤C(1 +|ξ|)^−A. (9) Then, for all k, n∈Z, and m, j ∈N₀, we have

|hψ_j,k, g_n,mi| ≤C˜·2^−j/2(1 + 2^−j|m|)^−A,

|hφk, gn,mi| ≤C˜·(1 +|m|)^−A, with

C˜ := (1−A⁻¹)⁻¹·2^3/2(2^A+ 3^A)C². (10)

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Lemma 2 is proved using the frequency localization of the wavelet and Wilson basis.

The next simple lemma uses the time localization to obtain other estimates of the inner products.

Lemma 3. Suppose that supp(φ),supp(ψ) ⊆ [−λ, λ] for some λ < ∞, and supp(g) ⊆ [−1/2,3/2]. Then we have the estimates

|hψj,k, gn,mi| ≤

(2^−j/2+1min{λ,2^j}, if 2^j(n−1/2)−λ < k < 2^j(n+ 3/2) +λ

0, else,

and

|hφk, gn,mi| ≤

(2λ, if n−1/2−λ < k < n+ 3/2 +λ 0, else.

for all j, k, n ∈Z, andm ∈N.

2.5 Upper bound on the cumulative coherence

Using the estimates above we can now upper bound the cumulative coherence as expressed in the following result.

Theorem 2. Let φ, ψ, and g be three univariate functions with supp(φ),supp(ψ) ⊆ [−λ, λ] for some λ <∞, and supp(g)⊆ [−1/2,3/2]. Assume that for some C <∞ and A >1 we have

max(|φ(ξ)b |,|ψ(ξ)b |,|bg(ξ)|)≤C(1 +|ξ|)^−A. (11) Consider a support set I which does not contain any scaling function, and no Wilson function gn,0 of frequency index 0 either. Then, for any p,

µ^p_p(Φ, I)≤2(λ+ 1)·C˜^p·X

j≥0

N_j(I)·2^−jp/2. (12) with C˜ given by (10).

Comparing this result with the sketch (8) we notice that, in addition to a constant factor, the supremum over j has been replaced by a sum, which is quite strong but does not fundamentally change the rate at which N_j(I) can grow with j. One can compare it to going from a weak ℓ1 norm to a strong ℓ1-norm.

Proof. First we estimate sup_g′∈F^g

P

ψ∈I^φ∪I^ψ|hg^′, ψi|^p. For that we consider a given Wilson basis function g^′ =gn^′,m^′ ∈F^g. By Lemma 2 we have, for all j ≥0 and k∈Z

|hg^′, ψ_j,ki| ≤C˜·2^−j/2(1 + 2^−j|m^′|)^−A.

By Lemma 3 the only indiceskwhich may yield a nonzero inner product are in the interval 2^j(n^′−¹₂)−λ,2^j(n^′+³₂) +λ

, which covers at most 2 + 2^1−jλ ≤2(λ+ 1) intervals of the form [2ⁿℓ,2^j^′(n+ 1)). Therefore, we may take thep-th power and sum up to get

X

n∈Z

X

ψj,k∈I_j,n^ψ

|hg_n,m, ψ_j,ki|^p ≤2(λ+ 1)·N_j(I)·C˜^p·2^−jp/2·(1 + 2^−j|m^′|)^−Ap.

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Since we assume I^ψ is empty, summing up over j gives X

ψ∈I^φ∪I^ψ

|hg^′, ψi|^p ≤2(λ+ 1)·C˜^p·X

j≥0

Nj(I)·2^−jp/2·(1 + 2^−j|m^′|)^−Ap

and by taking the supremum over m, which is achieved atm = 0, it follows that sup

g^′∈F^g

X

ψ∈I^ψ∪I^φ

|hg^′, ψi|^p ≤2(λ+ 1)·C˜^p·X

j≥0

2^−jp/2·Nj(I) (13) Reversing now the roles between Wilson basis functions and wavelets we now want to estimate sup_ψ′∈F^φ∪F^ψ

P

g∈I^g|hψ^′, gi|^p. We consider a given wavelet ψ^′ = ψj^′,k^′. By Lemma 2 we have for any n∈Z, m≥1 and j ≥0 such that 2^j ≤m <2^j+1

|hψ^′, gn,mi| ≤C˜·2^−j^′^/2(1 + 2^−j^′|m|)^−A≤C˜·2^−j^′^/2(1 + 2^−j^′2^j)^−A.

Moreover, by Lemma 3, the only indices n for which this inner product can be nonzero satisfy −¹₂ −2^−j^′λ < n−2^−j^′k^′ < ³₂ + 2^−j^′λ, so there is at most λ2^1−j^′+ 1 ≤2(λ+ 1) of them. Therefore, taking the p-th power and summing we get

X

n∈Z

X

gn,m∈I_j,n^g

|hψ^′, g_n,mi|^p ≤2(λ+ 1)·N_j(I)·C˜^p·2^−j^′^p/2·(1 + 2^j−j^′)^−Ap.

Since we assume I^g does not contain any gn,m with m = 0, summing up over j gives X

g∈I^g

|hψ^′, gi|^p ≤2(λ+ 1)·X

j≥0

N_j(I)·C˜^p·2^−j^′^p/2 ·(1 + 2^j−j^′)^−Ap (14)

Similarly, for any scaling function ψ^′ =φk we obtain X

g∈I^g

|hψ^′, gi|^p ≤2(λ+ 1)·X

j≥0

Nj(I)·C˜^p·(1 + 2^j)^−Ap

and we notice that the right hand side is exactly that of (14) for j^′ = 0. Therefore we obtain

sup

ψ^′∈F^φ∪F^ψ

X

g∈I^g

|hψ^′, gi|^p ≤2(λ+ 1)·C˜^p·sup

j^′≥0

X

j≥0

2^−j^′^p/2·(1 + 2^j−j^′)^−Ap·Nj(I)

. (15)

SinceA > ¹₂ it follows that for anyℓ∈Z, 2^ℓp/2(1 + 2^ℓ)^−Ap≤1. Therefore, for anyj, j^′ ≥0 we have

2^−j^′^p/2·(1 + 2^j−j^′)^−Ap = 2^(j−j^′^)p/2·(1 + 2^j−j^′)^−Ap·2^−jp/2 ≤2^−jp/2. Combining these facts with (13) and (15) we get the desired result (12).

We have the following corollary.

Irisa

(14)

Corollary 1. Assume that a wavelet basis and a Wilson basis satisfy the decay conditions of Theorem 2, and consider a support set I which does not contain any scaling function, and no Wilson function g_n,0 of frequency index 0 either. If

X

j≥0

N_j(I)·2^−j/2 < 1

4(λ+ 1) ˜C (16)

then all standard pursuit algorithms will (stably) recover the support of any combination of atoms from the support set I.

Proof. For any ℓ ∈ I we consider the subset Jℓ = I\{ℓ} and notice that, for all j, Nj(Jℓ)≤Nj(I), therefore, applying Theorem 2 withp= 1 we get under the condition (16) that

µ1(Φ, I) + sup

ℓ∈I

µ1(Φ_I, Jℓ)≤µ1(Φ, I) + sup

ℓ∈I

µ1(Φ, Jℓ)≤4(λ+ 1) ˜C·X

j≥0

Nj(I)·2^−j/2 <1.

Our theorem can also be combined with the main theorem in [9] to prove that, under a white Gaussian model on the coefficients S, if

X

j≥0

Nj(I)·2^−j < 1

2(λ+ 1) ˜C² (17)

then the probability that multichannel thresholding fails to recover the support set I decays exponentially fast with the number of channels.

Example 1. The compactly supported Daubechies wavelets{φ^N}, {ψ^N}(filter length 2N) satisfy supp(φ^N),supp(ψ^N)⊆[−N, N] with

max(|φc^N(ξ)|,|ψc^N(ξ)|)≤C(1 +|ξ|)^−µN−1,

with µ ≈ 0.1887, see [2, Chap. 7]. Thus we can apply Theorem 2 and its corollary with A=µN + 1 with any infinitely differentiable cut-off function g with supp(g)⊆[−¹₂,³₂].

3 Conclusion

In this note we have derived sufficient conditions for the Basis Pursuit and Matching Pursuit algorithms to recover structured representations of admissible signal with respect to an infinite dictionary given as the union of a nice wavelet basis and a Wilson basis.

The sufficient conditions, although quite natural given the known coherence results for finite dictionaries, take into account the time-frequency structure of the dictionary and are thus much more optimistic than estimates taking only into account the overall dictionary coherence or its cumulative coherence. The conditions allow very large (even infinite) support sets to be recovered. These results somehow explain the success of audio signal processing techniques such as those proposed by Daudet and Torr´esani [3] in recovering meaningful signal representations in a union of a wavelet and a local Fourier basis

PI n˚1833